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A049417 a(n) = isigma(n): sum of infinitary divisors of n. 84
1, 3, 4, 5, 6, 12, 8, 15, 10, 18, 12, 20, 14, 24, 24, 17, 18, 30, 20, 30, 32, 36, 24, 60, 26, 42, 40, 40, 30, 72, 32, 51, 48, 54, 48, 50, 38, 60, 56, 90, 42, 96, 44, 60, 60, 72, 48, 68, 50, 78, 72, 70, 54, 120, 72, 120, 80, 90, 60, 120, 62, 96, 80, 85, 84, 144, 68, 90 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
A divisor of n is called infinitary if it is a product of divisors of the form p^{y_a 2^a}, where p^y is a prime power dividing n and sum_a y_a 2^a is the binary representation of y.
Multiplicative: If e = Sum_{k >= 0} d_k 2^k (binary representation of e), then a(p^e) = Product_{k >= 0} (p^(2^k*{d_k+1}) - 1)/(p^(2^k) - 1). - Christian G. Bower and Mitch Harris, May 20 2005 [This means there is a factor p^2^k + 1 if d_k = 1, else the factor is 1. - M. F. Hasler, Oct 20 2022]
This sequence is an infinitary analog of the Dedekind psi function A001615. Indeed, a(n) = Product_{q in Q_n}(q+1) = n*Product_{q in Q_n} (1+1/q), where {q} are terms of A050376 and Q_n is the set of distinct q's whose product is n. - Vladimir Shevelev, Apr 01 2014
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..7417 from R. J. Mathar)
Graeme L. Cohen, On an integer's infinitary divisors, Math. Comp. 54 (189) (1990) 395-411.
Steven R. Finch, Unitarism and Infinitarism, February 25, 2004. [Cached copy, with permission of the author]
J. O. M. Pedersen, Tables of Aliquot Cycles [Broken link]
J. O. M. Pedersen, Tables of Aliquot Cycles [Via Internet Archive Wayback-Machine]
J. O. M. Pedersen, Tables of Aliquot Cycles [Cached copy, pdf file only]
Tomohiro Yamada, Infinitary superperfect numbers, arXiv:1705.10933 [math.NT], 2017.
FORMULA
Let n = Product(q_i) where {q_i} is a set of distinct terms of A050376. Then a(n) = Product(q_i + 1). - Vladimir Shevelev, Feb 19 2011
If n is squarefree, then a(n) = A001615(n). - Vladimir Shevelev, Apr 01 2014
a(n) = Sum_{k>=1} A077609(n,k). - R. J. Mathar, Oct 04 2017
a(n) = A126168(n)+n. - R. J. Mathar, Oct 05 2017
Multiplicative with a(p^e) = Product{k >= 0, e_k = 1} p^2^k + 1, where e = Sum e_k 2^k, i.e., e_k is bit k of e. - M. F. Hasler, Oct 20 2022
EXAMPLE
If n = 8: 8 = 2^3 = 2^"11" (writing 3 in binary) so the infinitary divisors are 2^"00" = 1, 2^"01" = 2, 2^"10" = 4 and 2^"11" = 8; so a(8) = 1+2+4+8 = 15.
n = 90 = 2*5*9, where 2, 5, 9 are in A050376; so a(n) = 3*6*10 = 180. - Vladimir Shevelev, Feb 19 2011
MAPLE
isidiv := proc(d, n)
local n2, d2, p, j;
if n mod d <> 0 then
return false;
end if;
for p in numtheory[factorset](n) do
padic[ordp](n, p) ;
n2 := convert(%, base, 2) ;
padic[ordp](d, p) ;
d2 := convert(%, base, 2) ;
for j from 1 to nops(d2) do
if op(j, n2) = 0 and op(j, d2) <> 0 then
return false;
end if;
end do:
end do;
return true;
end proc:
idivisors := proc(n)
local a, d;
a := {} ;
for d in numtheory[divisors](n) do
if isidiv(d, n) then
a := a union {d} ;
end if;
end do:
a ;
end proc:
A049417 := proc(n)
local d;
add(d, d=idivisors(n)) ;
end proc:
seq(A049417(n), n=1..100) ; # R. J. Mathar, Feb 19 2011
MATHEMATICA
bitty[k_] := Union[Flatten[Outer[Plus, Sequence @@ ({0, #1} & ) /@ Union[2^Range[0, Floor[Log[2, k]]]*Reverse[IntegerDigits[k, 2]]]]]]; Table[Plus@@((Times @@ (First[it]^(#1 /. z -> List)) & ) /@ Flatten[Outer[z, Sequence @@ bitty /@ Last[it = Transpose[FactorInteger[k]]], 1]]), {k, 2, 120}]
(* Second program: *)
a[n_] := If[n == 1, 1, Sort @ Flatten @ Outer[ Times, Sequence @@ (FactorInteger[n] /. {p_, m_Integer} :> p^Select[Range[0, m], BitOr[m, #] == m &])]] // Total;
Array[a, 100] (* Jean-François Alcover, Mar 23 2020, after Paul Abbott in A077609 *)
PROG
(PARI) A049417(n) = {my(b, f=factorint(n)); prod(k=1, #f[, 2], b = binary(f[k, 2]); prod(j=1, #b, if(b[j], 1+f[k, 1]^(2^(#b-j)), 1)))} \\ Andrew Lelechenko, Apr 22 2014
(PARI) isigma(n)=vecprod([vecprod([f[1]^2^k+1|k<-[0..exponent(f[2])], bittest(f[2], k)])|f<-factor(n)~]) \\ M. F. Hasler, Oct 20 2022
(Haskell)
a049417 1 = 1
a049417 n = product $ zipWith f (a027748_row n) (a124010_row n) where
f p e = product $ zipWith div
(map (subtract 1 . (p ^)) $
zipWith (*) a000079_list $ map (+ 1) $ a030308_row e)
(map (subtract 1 . (p ^)) a000079_list)
-- Reinhard Zumkeller, Sep 18 2015
CROSSREFS
Cf. A049418 (3-infinitary), A074847 (4-infinitary), A097863 (5-infinitary).
Sequence in context: A322485 A327668 A324706 * A331110 A188999 A186644
KEYWORD
nonn,mult
AUTHOR
Yasutoshi Kohmoto, Dec 11 1999
EXTENSIONS
More terms from Wouter Meeussen, Sep 02 2001
STATUS
approved

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Last modified March 28 15:58 EDT 2024. Contains 371254 sequences. (Running on oeis4.)